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Finding the linear transformation of a matrix

  1. Dec 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Hello again. First of all thanks to anyone who has replied to my previous questions.
    Now, the question that troubles me is:

    We are given a matrix A2x2 with some random values and we are asked to say if there is a linear map which has A as its map for the standar basis.

    A= 2 1
    -5 8

    2. Relevant equations

    3. The attempt at a solution
    If A is the matrix of some linear map for the standar basis that means that:

    f(1,0)=2(1,0) -5(0,1)
    f(0,1)=1(1,0) +8(0,1)

    so f(1,0)=(2,-5) and f(0,1)=(1,8)

    Although the question only asks to say if such a linear map exists (it obviously does but i dont know how to prove it) I would be glad if someone could instruct me on finding formulas for linear maps when we have their matrix.
  2. jcsd
  3. Dec 12, 2009 #2


    Staff: Mentor

    So given a matrix, you are trying to find the linear transformation represented by the matrix. Building on the work you already have, and identifying the transformation by T, we have.

    [tex]T\left(\begin{array}{c}x&y\end{array} \right)[/tex]
    [tex]=~x~T\left(\begin{array}{c}1&0\end{array} \right)~+~y~T\left(\begin{array}{c}0&1\end{array} \right)[/tex]
    [tex]=~x~\left(\begin{array}{c}2&-5\end{array} \right)~+~y~T\left(\begin{array}{c}1&8\end{array} \right)[/tex]
    [tex]=~\left[\begin{array}{cc}?&?&?&?\end{array} \right]~\left(\begin{array}{c}x&y\end{array} \right)[/tex]

    Can you fill in the missing entries in the 2 x 2 matrix?
  4. Dec 12, 2009 #3
    Yes I think I got it now. The missing matrix is the matrix i posted on my first post. So T(x,y)=(2x+y,-5x+8y). The way you put it, I see that a matrix and a linear transformation are actually the exact same thing, expressed in a different way? Thank you very much, the work you people do here is incredible.
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